Upscaling and Inverse Modeling of Groundwater Flow and Mass ...

10 CHAPTER 2. A COMPARATIVE STUDY OF THREE- . . .
Table 2.1: Variogram parameters for the model fit in Figure 2.2
Model Type Sill Range [m]
c ax ay az
Nugget 0.424
Spherical 3.820 32 80 4.1
Hole effect 0.891 ∞ 80 ∞
where h = (hx, hy, hz) is the separation vector, ax1 , ay1 , az1 are the ranges
of the spherical variogram, ax2 , ay2 , az2 are the ranges of the hole-effect variogram,
∥ · ∥ denotes vector modulus, c0 is the nugget, c1 is the sill of the
spherical model, c2 is the sill of the hole-effect model, with the y-axis oriented
parallel to the flow direction, the x-axis is orthogonal to it on the horizontal
plane, and the z-axis is parallel to the vertical direction. The parameter values
used to fit the experimental variogram are given in Table 2.1. Notice that ay2 ,
and az2 are equal to infinity, meaning that the hole-effect is only present along
the flow direction. The fitted model is also shown in Figure 2.2.
Figure 2.2: Horizontal and vertical experimental variograms, and fitted model,
for the lnK flowmeter data. The rotation angle of the directional variograms
is measured in degrees clockwise from the positive y-axis.